Derivative of e^x²

We aim to understand the derivative of e^(x²). It is commonly represented as d/dx (e^(x²)) or (e^(x²))', and its value is 2x·e^(x²).

The function e^(x²) has a clear derivative, indicating that it is differentiable over its entire domain. Key concepts include:

Exponential Function: e^(x²), where e is the base of natural logarithms.

Chain Rule: A rule for differentiating composite functions, essential for e^(x²).

Product Rule: Used in some methods for differentiation, though not directly for e^(x²).

The derivative of e^(x²) is denoted as d/dx (e^(x²)) or (e^(x²))'.

The formula used to differentiate e^(x²) is: d/dx (e^(x²)) = 2x·e^(x²)

This formula applies to all x in the real number domain, as e^(x²) is defined everywhere.

The derivative of e^(x²) can be derived using different methods. Here, we'll use the chain rule and the first principle to demonstrate it:

Using Chain Rule

To prove the differentiation of e^(x²) using the chain rule: Let u = x², then the function becomes e^u. The derivative of e^u with respect to u is e^u. The derivative of u = x² with respect to x is 2x.

By the chain rule, d/dx (e^(x²)) = e^u · du/dx = e^(x²) · 2x = 2x·e^(x²).

By First Principle

We can also derive the derivative using the first principle, which expresses the derivative as the limit of the difference quotient. Let f(x) = e^(x²).

Its derivative can be expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [e^((x+h)²) - e^(x²)] / h = limₕ→₀ [e^(x² + 2xh + h²) - e^(x²)] / h = limₕ→₀ [e^(x²) · (e^(2xh + h²) - 1)] / h

Using the expansion of e^u for small u, e^u ≈ 1 + u, we have: = limₕ→₀ [e^(x²) · (2xh + h²)] / h = limₕ→₀ [e^(x²) · (2x + h)] = e^(x²) · 2x

Hence, the derivative of e^(x²) is 2x·e^(x²).

When a function is differentiated multiple times, the subsequent derivatives are called higher-order derivatives. Understanding these derivatives provides deeper insights into the function's behavior.

For the first derivative of a function, we denote it as f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, denoted as f′′(x), which shows the rate of change of the rate of change. Similarly, the third derivative, f′′′(x), results from the second derivative, and this pattern continues.

For the nth derivative of e^(x²), we generally use fⁿ(x) to denote the nth derivative of a function f(x), which tells us about the change in the rate of change, continuing for higher-order derivatives.

Since e^(x²) is defined for all real x, there are no points where the derivative is undefined.

At x = 0, the derivative of e^(x²) = 2x·e^(x²) simplifies to 0·e^(0) = 0, indicating no change at that point.

• Derivative: The derivative of a function indicates how the function changes in response to a slight change in x.

• Exponential Function: A function in which an independent variable appears in the exponent, such as e^(x²).

• Chain Rule: A differentiation rule used for computing the derivative of the composition of two or more functions.

• Product Rule: A differentiation rule used when differentiating the product of two functions.

• Quotient Rule: A differentiation rule used when differentiating the quotient of two functions.